Duck egg pros: They are better in every respect.
Duck egg cons: None.
I simply will not go back to chicken eggs. Ever, if I can help it. My friend who is a chef buys my eggs and she is always so grateful, especially if it's been a while since we've been able to get up with each other and she's had to resort to chicken eggs.
I never buy eggs in a restaurant. Ick.
I never buy chicken eggs, period, ever.
Okay, in all seriousness and bias aside:
Duck eggs are higher in protein, vitamins, minerals, and other nutrients. Also higher in cholesterol and fat, although new research is showing that those nutrients can be healthy too when they come from a healthy source (like fresh, free ranged duck eggs, for instance).
Duck eggs have a slightly firmer texture than chicken eggs that may be a little disconcerting for people accustomed to chicken eggs. Add a little milk when scrambling to soften the texture. Personally, I like the firmer texture.
FRESH, free-range eggs of any sort will taste richer and be darker in color, both of which qualities indicate higher levels of nutrients.
Duck eggs will "stand up" in the pan better than chicken eggs.
Duck egg shells are hard to crack and until you get the hang of it, you're likel to have shells in your eggs occasionally.
Very fresh eggs are harder to peel when boiled.
Duck eggs contain different proteins from chicken eggs. Some people who are allergic to chicken eggs can eat duck eggs just fine. Likewise, some people who can eat chicken eggs just fine may be allergic to duck eggs. A mild allergy is usually the reason for the "overly full" feeling someone described. Some friends of mine get nauseous from duck eggs, and one person in particular ends up in bed for half a day if he eats any. This is unusual, and not dangerous--egg allergies take time to build up, so you won't keel over on your first attempt, even if you are allergic. You just might get a little sick. But isn't it nice that people who are allergic to one kind of eggs can often still eat the other kind?
Duck eggs are categorically better for baking. Thanks to their different proteins, they provide more "loft" and better moisture. You can use fewer to get the same effect, or use the number called for in the recipe to get exceptionally moist and lofty baked goods. There will be no difference in flavor.
I agree that in taste tests, it is a rare person who could tell the difference. They really do taste basically the same with minor texture differences.
However, free range eggs--duck or chicken or any other sort--will occasionally carry flavors from something the animal has been eating. Every once in a while, I'll crack an egg open and get a slightly fishy or onion-pungent odor. Usually, the eggs taste fine once cooked, and the flavor is very mild. I have heard that ducks with a heavy fish diet will develop that odor in their eggs. I think the wild onions they eat in Spring probably have an effect, too. It is unusual for me to have this problem, and the eggs are still perfectly usable, but it does happen. Usually, it's only two or three eggs at a time, and it only happens two or three times a year. I suspect that chicken eggs can occasionally have the issue too--it's the difference between factory produced eggs from hens who all eat exactly the same thing every day with never access to anything fresh and whose eggs are going to always be exactly the same--pale, flavorless, and runny--versus eggs from hens who forage bugs and grass and fish and worms all day and whose eggs taste fresh and clean and rich and, occasionally, tinged with an odor of what they've been eating.
Hope that helps.
It's a normal reflex in humans to be wary of new foods. I feed duck eggs to anyone who comes over & never bother telling them what kind of eggs they are. Some folks already know, some folks don't care, others might turn their noses up if they knew, but whatever. I do know who among my friends is allergic to duck eggs, and of COURSE I never serve them eggs. I think the best thing a person can do to encourage friends and family to eat duck eggs is simply to eat and manifestly ENJOY the eggs yourself. They'll come around.